Group Theory in the Bedroom

The absence of a golden rule for mattress flipping is a
disappointment, but it does not portend the demise of Western
Civilization. We can adapt; we can learn to live with it.

Suppose you flip your mattress at regular intervals, but each time
you choose an axis of rotation at random. This is clearly a
less-than-optimal algorithm, but how large a penalty does it carry?
Under the ideal rotation schedule, each orientation of the mattress
would get 25 percent of the wear. A quick computer simulation shows
that if you do random flips quarterly over a period of 10 years, the
most-used orientation will get 31 percent of the wear and the
least-used 19 percent. Except for those among us who suffer from a
severe Princess-and-the-Pea complex, ±6 percent is probably
good enough.

But we can do even better. We can cheat.

The no-golden-rule theorem for mattress flipping assumes that the
surfaces of the mattress are unmarked, so that the four allowed
configurations are indistinguishable. Everything changes if you
label the surfaces. Specifically, suppose you go through the four
possible orientations of the mattress and label each one with a
number from the set 0, 1, 2, 3. You might place the labels so that
in each configuration, one of these numbers is facing upward in the
corner closest to the righthand side of the headboard. Given this
labeling, the mattress-flipping algorithm calls for nothing more
than counting. Each time you are ready to make a flip, you note the
number that appears in the upper righthand corner, and calculate the
successor of that number modulo 4. (In other words, you cycle
through the sequence 0, 1, 2, 3 and then return to 0 again.) Turn
the mattress in whatever way is necessary to bring the successor
number into the upper-right position. The turn needed will not
always be around the same axis, but the closure property of the
group guarantees that you will always be able to bring the next
number into position with a single flip around some axis.

The counting algorithm is not a golden rule, but perhaps it
deserves to be called a silver one. As a practical matter, this
solution is so simple that I would expect mattress makers to adopt
it, by embossing numbers on their products. Some of the
manufacturers require periodic flipping as a condition of
maintaining a warranty, and they give complicated—often
ambiguous—instructions on how to comply. Wouldn't it be easier
just to count?

The algorithm can be simplified even further for those who can't
count as high as 3. If a mattress had lengthwise stripes on one side
and crosswise stripes on the other, you could cycle through the four
states by always flipping parallel to the stripes. Another
possibility is to somehow adapt to our purposes the label that reads
"Do not remove this label." (Now there's a golden rule!)